de.jtem.ellipticFunctions

Class Elliptic

java.lang.Object

de.jtem.ellipticFunctions.Elliptic

Direct Known Subclasses:

Jacobi

public class Elliptic
extends Object

Class provides Jacobi's elliptic theta functions.

Author:

Markus Schmies, Boris Springborn

Method Summary

Modifier and Type	Method and Description
static de.jtem.mfc.field.Complex	theta(de.jtem.mfc.field.Complex z, de.jtem.mfc.field.Complex tau) Computes θ(z,τ).
static void	theta(de.jtem.mfc.field.Complex w, de.jtem.mfc.field.Complex tau, de.jtem.mfc.field.Complex z, de.jtem.mfc.field.Complex logOfC, de.jtem.mfc.field.Complex thetaOfZ) Helps to compute θ(w,τ) = c θ(z,τ).
static de.jtem.mfc.field.Complex	theta0(de.jtem.mfc.field.Complex z, de.jtem.mfc.field.Complex tau) Computes θ0(z,τ).
static void	theta0(de.jtem.mfc.field.Complex w, de.jtem.mfc.field.Complex tau, de.jtem.mfc.field.Complex z, de.jtem.mfc.field.Complex logOfC, de.jtem.mfc.field.Complex thetaOfZ) Helps to compute θ0(w,τ) = c θ(z,τ).
static de.jtem.mfc.field.Complex	theta1(de.jtem.mfc.field.Complex z, de.jtem.mfc.field.Complex tau) Computes θ1(z,τ).
static void	theta1(de.jtem.mfc.field.Complex w, de.jtem.mfc.field.Complex tau, de.jtem.mfc.field.Complex z, de.jtem.mfc.field.Complex logOfC, de.jtem.mfc.field.Complex thetaOfZ) Helps to compute θ1(w,τ) = c θ(z,τ).
static de.jtem.mfc.field.Complex	theta2(de.jtem.mfc.field.Complex z, de.jtem.mfc.field.Complex tau) Computes θ2(z,τ).
static void	theta2(de.jtem.mfc.field.Complex w, de.jtem.mfc.field.Complex tau, de.jtem.mfc.field.Complex z, de.jtem.mfc.field.Complex logOfC, de.jtem.mfc.field.Complex thetaOfZ) Helps to compute θ2(w,τ) = c θ(z,τ).
static de.jtem.mfc.field.Complex	theta3(de.jtem.mfc.field.Complex z, de.jtem.mfc.field.Complex tau) Computes θ3(z,τ).
static void	theta3(de.jtem.mfc.field.Complex w, de.jtem.mfc.field.Complex tau, de.jtem.mfc.field.Complex z, de.jtem.mfc.field.Complex logOfC, de.jtem.mfc.field.Complex thetaOfZ) Helps to compute θ3(w,τ) = c θ(z,τ).
static de.jtem.mfc.field.Complex	theta4(de.jtem.mfc.field.Complex z, de.jtem.mfc.field.Complex tau) Computes θ4(w,τ).
static void	theta4(de.jtem.mfc.field.Complex w, de.jtem.mfc.field.Complex tau, de.jtem.mfc.field.Complex z, de.jtem.mfc.field.Complex logOfC, de.jtem.mfc.field.Complex thetaOfZ) Helps to compute θ4(w,τ) = c θ(z,τ).
static de.jtem.mfc.field.Complex[]	thetaConstants(de.jtem.mfc.field.Complex tau) Returns theta constants θ0(0,τ), θ1(0,τ) = 0, θ2(0,τ) = 0, and θ3(0,τ).
static void	thetaConstants(de.jtem.mfc.field.Complex tau, de.jtem.mfc.field.Complex[] c) Computes theta constants θ0(0,τ), θ1(0,τ) = 0, θ2(0,τ) = 0, and θ3(0,τ).

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

theta

public static de.jtem.mfc.field.Complex theta(de.jtem.mfc.field.Complex z,
                                              de.jtem.mfc.field.Complex tau)

Computes θ(z,τ). θ, θ₀, θ₄ refer to the same function, thus theta(Complex,Complex) , theta0(Complex,Complex), and theta4(Complex,Complex) are identical.

Parameters:

z - argument

tau - lattice parameter

Returns:

θ(z,τ)

See Also:

theta(Complex,Complex,Complex,Complex,Complex)

theta

public static void theta(de.jtem.mfc.field.Complex w,
                         de.jtem.mfc.field.Complex tau,
                         de.jtem.mfc.field.Complex z,
                         de.jtem.mfc.field.Complex logOfC,
                         de.jtem.mfc.field.Complex thetaOfZ)

Helps to compute θ(w,τ) = c θ(z,τ). θ, θ₀, θ₄ refer to the same function, thus theta(Complex,Complex,Complex,Complex,Complex), theta0(Complex,Complex,Complex,Complex,Complex), and theta4(Complex,Complex,Complex,Complex,Complex) are idenical.

Parameters:

w - argument

tau - lattice parameter

z - = x π + y π τ with -0.5 <= x,y <= 0.5 (on output)

logOfC - equals log(c) (on output)

thetaOfZ - equals θ(z,τ) (on output)

See Also:

theta(Complex,Complex)

theta0

public static de.jtem.mfc.field.Complex theta0(de.jtem.mfc.field.Complex z,
                                               de.jtem.mfc.field.Complex tau)

Computes θ₀(z,τ). θ, θ₀, θ₄ refer to the same function, thus theta(Complex,Complex), theta0(Complex,Complex), and theta4(Complex,Complex) are identical.

Parameters:

z - argument

tau - lattice parameter

Returns:

θ₀(z,τ)

See Also:

theta0(Complex,Complex,Complex,Complex,Complex)

theta0

public static void theta0(de.jtem.mfc.field.Complex w,
                          de.jtem.mfc.field.Complex tau,
                          de.jtem.mfc.field.Complex z,
                          de.jtem.mfc.field.Complex logOfC,
                          de.jtem.mfc.field.Complex thetaOfZ)

Helps to compute θ₀(w,τ) = c θ(z,τ). θ, θ₀, θ₄ refer to the same function, thus theta(Complex,Complex,Complex,Complex,Complex), theta0(Complex,Complex,Complex,Complex,Complex), and theta4(Complex,Complex,Complex,Complex,Complex) are identical.

Parameters:

w - argument

tau - lattice parameter

z - = x π + y π τ with -0.5 <= x,y <= 0.5 (on output)

logOfC - equals log(c) (on output)

thetaOfZ - equals θ(z,τ) (on output)

See Also:

theta0(Complex,Complex)

theta1

public static de.jtem.mfc.field.Complex theta1(de.jtem.mfc.field.Complex z,
                                               de.jtem.mfc.field.Complex tau)

Computes θ₁(z,τ).

Parameters:

z - argument

tau - lattice parameter

Returns:

θ₁(z,τ)

See Also:

theta1(Complex,Complex,Complex,Complex,Complex)

theta1

public static void theta1(de.jtem.mfc.field.Complex w,
                          de.jtem.mfc.field.Complex tau,
                          de.jtem.mfc.field.Complex z,
                          de.jtem.mfc.field.Complex logOfC,
                          de.jtem.mfc.field.Complex thetaOfZ)

Helps to compute θ₁(w,τ) = c θ(z,τ).

Parameters:

w - argument

tau - lattice parameter

z - equals x π + y π τ with -0.5 <= x,y <= 0.5 (on output)

logOfC - equals log(c) (on output)

thetaOfZ - equals θ(z,τ) (on output)

See Also:

theta1(Complex,Complex)

theta2

public static de.jtem.mfc.field.Complex theta2(de.jtem.mfc.field.Complex z,
                                               de.jtem.mfc.field.Complex tau)

Computes θ₂(z,τ).

Parameters:

z - argument

tau - lattice parameter

Returns:

θ₂(z,τ)

See Also:

theta2(Complex,Complex,Complex,Complex,Complex)

theta2

public static void theta2(de.jtem.mfc.field.Complex w,
                          de.jtem.mfc.field.Complex tau,
                          de.jtem.mfc.field.Complex z,
                          de.jtem.mfc.field.Complex logOfC,
                          de.jtem.mfc.field.Complex thetaOfZ)

Helps to compute θ₂(w,τ) = c θ(z,τ).

Parameters:

w - argument

tau - lattice parameter

z - equals x π + y π τ with -0.5 <= x,y <= 0.5 (on output)

logOfC - equals log(c) (on output)

thetaOfZ - equals θ(z,τ) (on output)

See Also:

theta2(Complex,Complex)

theta3

public static de.jtem.mfc.field.Complex theta3(de.jtem.mfc.field.Complex z,
                                               de.jtem.mfc.field.Complex tau)

Computes θ₃(z,τ).

Parameters:

z - argument

tau - lattice parameter

Returns:

θ₃(z,τ)

See Also:

theta3(Complex,Complex,Complex,Complex,Complex)

theta3

public static void theta3(de.jtem.mfc.field.Complex w,
                          de.jtem.mfc.field.Complex tau,
                          de.jtem.mfc.field.Complex z,
                          de.jtem.mfc.field.Complex logOfC,
                          de.jtem.mfc.field.Complex thetaOfZ)

Helps to compute θ₃(w,τ) = c θ(z,τ).

Parameters:

w - argument

tau - lattice parameter

z - equals x π + y π τ with -0.5 <= x,y <= 0.5 (on output)

logOfC - equals log(c) (on output)

thetaOfZ - equals θ(z,τ) (on output)

See Also:

theta3(Complex,Complex)

theta4

public static de.jtem.mfc.field.Complex theta4(de.jtem.mfc.field.Complex z,
                                               de.jtem.mfc.field.Complex tau)

Computes θ₄(w,τ). θ, θ₀, θ₄ refer to the same function, thus theta(Complex,Complex), theta0(Complex,Complex), and theta4(Complex,Complex) are identical.

Parameters:

z - argument

tau - lattice parameter

See Also:

theta4(Complex,Complex,Complex,Complex,Complex)

theta4

public static void theta4(de.jtem.mfc.field.Complex w,
                          de.jtem.mfc.field.Complex tau,
                          de.jtem.mfc.field.Complex z,
                          de.jtem.mfc.field.Complex logOfC,
                          de.jtem.mfc.field.Complex thetaOfZ)

Helps to compute θ₄(w,τ) = c θ(z,τ). θ, θ₀, θ₄ refer to the same function, thus theta(Complex,Complex,Complex,Complex,Complex), theta0(Complex,Complex,Complex,Complex,Complex), and theta4(Complex,Complex,Complex,Complex,Complex) are identical.

Parameters:

w - argument

tau - lattice parameter

z - = x π + y π τ with -0.5 <= x,y <= 0.5 (on output)

logOfC - =log(c) (on output)

thetaOfZ - = θ₄(z,τ) (on output)

See Also:

theta4(Complex,Complex)

thetaConstants

public static de.jtem.mfc.field.Complex[] thetaConstants(de.jtem.mfc.field.Complex tau)

Returns theta constants θ₀(0,τ), θ₁(0,τ) = 0, θ₂(0,τ) = 0, and θ₃(0,τ). The result is stored in the array c. The array may have null entries, which causes the routine to create instances.

Parameters:

tau - lattice paramter τ

Returns:

array with the four theta constants (on output)

thetaConstants

public static void thetaConstants(de.jtem.mfc.field.Complex tau,
                                  de.jtem.mfc.field.Complex[] c)

Computes theta constants θ₀(0,τ), θ₁(0,τ) = 0, θ₂(0,τ) = 0, and θ₃(0,τ). The result is stored in the array c. The array may have null entries, which causes the routine to create instances.

Parameters:

tau - lattice paramter τ

c - array with the four theta constants (on output)